Basic Concepts of Set Theory Functions The Natural Numbers and Mathematical Induction Ordered Field Axioms The Completeness Axiom for the Real Numbers Applications of the Completeness Axiom 1. TOOLS FOR ANALYSIS This chapter discusses various mathematical concepts and constructions which are central to the study of many fundamental results in analysis. Generalities are kept to a minimum in order to move quickly to the heart of analysis: the structure of the real number system and the notion of limit. The reader should consult the bibliographical references for more details. 1.1 Basic Concepts of Set Theory Intuitively, a set is a collection of objects with certain properties. The objects in a set are called the elements or members of the set. We usually use uppercase letters to denote sets and lowercase letters to denote elements of sets. If ais an element of a set A, we write a∈A. If ais not an element of a set A, we write a̸∈A. To specify a set, we can list all of its elements, if possible, or we can use a defining rule. For instance, to specify the fact that a set Acontains four elements a,b,c,d, we write A={a,b,c,d}. To describe the set E containing all even integers, we write E={x : x =2k for some integer k}. We say that a set Ais a subset of a set Bif every element of Ais also an element of B, and write A⊂Bor B⊃A. Two sets are equal if they contain the same elements. If Aand Bare equal, we write A=B. The following result is straightforward and very convenient for proving equality between sets. Theorem 1.1.1 Two sets AandBare equal if and only if A⊂BandB⊂A. If A⊂BandAdoes not equal B, we say that Ais a proper subset of B, and write A⊊B.
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